In 1923, J. L. Walsh investigated a set of functions which can be employed in waveform synthesis schemes analogous to the Fourier synthesis methods employing sines and cosines. The Walsh waveforms are binary valued sequences of bits which repeat over a basic interval. He published a paper entitled "A Closed Set of Normal Orthogonal Functions" describing these functions in the American Journal of Mathematics, vol. 45, p. 5 (1925). More recently, R. B. Lackey described Walsh functions in a paper entitled "The Wonderful World of Walsh Functions" presented at the Walsh Functions Symposium and published in the 1972 Proceedings, "Applications of Walsh Functions." AD-744 650. In a set of Walsh harmonics, or functions, there is a pair of rectangular waveforms, or bit patterns, for each harmonic. Each such pair of waveforms has the same bit sequence, but they are displaced from each other in analogous fashion to the sine and cosine components of a Fourier series. Any continuous repetitive waveform can be synthesized by addition of Walsh harmonics in appropriate proportions in analogous fashion to synthesis with Fourier components. Synthesis using Walsh functions appears to be more attractive than using Fourier components due to the greater ease of generating the Walsh functions with digital circuits than the Fourier harmonic components. Despite this apparent advantage, additive synthesis using Walsh functions has not found favor in musical instruments. One of the principle objections to this approach is that the coefficients for the Walsh harmonics cannot be selected independently, because each harmonic contains undesirably large amounts of many Fourier harmonics. Consequently, a simultaneous solution for all of the Walsh coefficients is required. This is a chore for a computer to do rather than a musician.
An additive synthesis system of the Fourier series type, using digital waveforms for each harmonic component, was disclosed by the present inventor in U.S. Pat. No. 4,070,943, entitled "Improved Organ Keying System", issued Jan. 31, 1978. The circuit arrangement used to generate the harmonic waveforms comprised a resistive path between a square wave tone source, in the form of a binary divider, and a tone utilization circuit; and a transistor switch connected in the resistive path so as to increase the absolute value of tone current during the second and third quarters of each half cycle. The transistor switch was operated by an exclusive-or gate having its inputs driven by divider stages one and two octaves above that used as the square wave tone source. By appropriate choice of the relative magnitudes of the steps in the resulting waveform, the third and fifth harmonics were effectively eliminated. The remaining high order harmonics were effectively eliminated by a low pass filter, or integrator, in the utilization circuit. One binary divider, or counter, was provided for the harmonics of order 0.5.times.2.sup.n, where n is an integer; a second was provided for harmonics of order 1.5.times.2.sup.n ; and a third was provided for the fifth harmonic. A rate multiplier was used to establish the required clock rates for the three dividers. The resulting harmonic components were essentially pure sine waves, whereby additive synthesis in the classical Fourier manner using a single control to select the desired strength, or coefficient, of each harmonic was provided.